Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=\frac{1}{x^{2}+1} $$

Short answer

The function is symmetric about the y-axis, no x-intercepts, y-intercept (0,1).

Step-by-step solution

Understanding the Equation

The given equation is \( y = \frac{1}{x^2 + 1} \). This is a rational function where the numerator is a constant and the denominator is a quadratic expression. We will analyze the function for symmetries and intercepts.

Check Symmetry

To check for symmetry, evaluate the function for \(-x\). \( y = \frac{1}{(-x)^2 + 1} = \frac{1}{x^2 + 1} \), which is the same as the original. Therefore, the function is symmetric about the y-axis.

Find x-Intercepts

To find the x-intercepts, set \( y = 0 \) and solve for \( x \): \( \frac{1}{x^2 + 1} = 0 \). Since the numerator \( 1 \) cannot be zero, there are no x-intercepts.

Find y-Intercept

To find the y-intercept, set \( x = 0 \): \( y = \frac{1}{0^2 + 1} = \frac{1}{1} = 1 \). The y-intercept is at (0, 1).

Plot the Graph

The graph is symmetric about the y-axis, has no x-intercepts, and a y-intercept at (0,1). The function decreases as \( |x| \) increases because \( x^2 + 1 \) grows, making \( \frac{1}{x^2 + 1} \) smaller. Plot points around these characteristics to sketch the curve.

Key concepts

Symmetry in Functions

Symmetry in functions is a useful concept in calculus when analyzing graphs. A function can exhibit different types of symmetry, such as symmetry about the y-axis, x-axis, or the origin.

In our case, we use the function \( y = \frac{1}{x^2 + 1} \). To determine if a function is symmetric about the y-axis, substitute \(-x\) for \(x\) in the original function and see if the expression remains unchanged:
\[ y = \frac{1}{(-x)^2 + 1} = \frac{1}{x^2 + 1} \]
This result is identical to the original function, confirming that it is symmetric about the y-axis. Y-axis symmetry implies that if you fold the graph along the y-axis, the two halves match perfectly.

Recognizing symmetry not only helps in plotting the graph but also simplifies calculations and predictions about the behavior of the function on the coordinate plane.

Intercepts in Graphs

Intercepts are critical points where a graph crosses the axes. For any function, the x-intercept occurs where the graph intersects the x-axis, and y-intercepts occur where the graph intersects the y-axis.

For the rational function \( y = \frac{1}{x^2 + 1} \):

• **X-intercepts**: Set \( y = 0 \) to find where the graph intersects the x-axis.
  \[ \frac{1}{x^2 + 1} = 0 \]The numerator is 1, which cannot be zero, so this function has no x-intercepts.


• **Y-intercepts**: Set \( x = 0 \) to find where the graph intersects the y-axis:
  \[ y = \frac{1}{0^2 + 1} = 1 \]This gives the point (0, 1) as the y-intercept.

Knowing where these intercepts occur is essential for sketching accurate graphs, providing starting points and guides for the graph's behavior across the axes.

Rational Functions in Calculus

Rational functions are fractions where the numerator and denominator are both polynomials. In calculus, understanding these functions is crucial as they appear frequently in various contexts, such as analyzing limits, derivatives, and integrals.

Let's break down the function \( y = \frac{1}{x^2 + 1} \):
The numerator is a constant (1), and the denominator is a simple quadratic polynomial \(x^2 + 1\). This structure leads to several important properties:

• **Continuous Nature**: The denominator \(x^2 + 1\) is never zero for real numbers, ensuring the function is continuous over all real numbers. There are no asymptotes, a common characteristic in rational functions.


• **Monotonic Behavior**: As \(|x|\) becomes larger, \(x^2 + 1\) increases, thus making \(\frac{1}{x^2 + 1}\) decrease. This indicates the function is decreasing as the distance from the origin increases.

Understanding these points makes graphing and analyzing rational functions more intuitive, highlighting how such functions behave across ranges of \(x\).